Chapter 15 — Congruence and Regular Polygons

Standard: 6.MG.4 — The student will determine congruence of segments, angles, and polygons.

By the end of this chapter you will be able to:

• Identify regular polygons (6.MG.4a)
• Draw lines of symmetry to divide regular polygons into two congruent parts (6.MG.4b)
• Determine the congruence of segments, angles, and polygons given their properties (6.MG.4c)
• Determine whether polygons are congruent or noncongruent according to the measures of their sides and angles (6.MG.4d)

Lessons: 15.1 Congruent Segments and Angles · 15.2 Regular Polygons · 15.3 Lines of Symmetry · 15.4 Congruent and Noncongruent Polygons

Lesson 15.1 — Congruent Segments and Angles

What congruent means

Two figures are congruent if they have the same size and the same shape. One could be placed exactly on top of the other and match perfectly — even if you have to slide, turn, or flip it to get there.

That definition sounds visual, but for the two simplest figures it becomes a measurement you can check.

• Two segments are congruent when they have the same length.
• Two angles are congruent when they have the same measure.

Nothing else matters. A segment's position on the page, the direction it points, whether it is drawn thick or thin — none of that changes its length, so none of it changes congruence.

Segments $\overline{AB}$ and $\overline{CD}$ above are congruent because both measure $3$ cm, even though one is horizontal and the other is slanted. Segment $\overline{EF}$ is not congruent to them, because $5 \ne 3$.

Notation you need to read and write

Geometry is picky about notation here, and for a good reason: it separates a figure from a measurement.

So these two statements say the same thing in two correct ways:

$$\overline{AB} \cong \overline{CD} \qquad \text{and} \qquad AB = CD$$

The first says the two segments are congruent figures. The second says their lengths are equal numbers. Likewise, $\angle P \cong \angle Q$ means exactly $m\angle P = m\angle Q$.

The rule of thumb. Use $\cong$ when you are talking about shapes and $=$ when you are talking about numbers. Writing $\overline{AB} = 5$ cm mixes the two, because a segment is a figure and $5$ cm is a number. Write $AB = 5$ cm instead.

To say two figures are not congruent, write $\ncong$: $\angle A \ncong \angle B$.

Angle measure does not depend on side length

This one surprises people. An angle is the amount of turn between two rays, and the rays go on as far as you like. Drawing them longer does not open the angle any wider.

Both angles above measure $45°$, so $\angle P \cong \angle Q$. The longer rays on the right make the drawing bigger, not the angle.

Marks on figures

Drawings show congruence with marks instead of numbers, which is handy when a problem does not give measurements.

• Tick marks on segments: segments with the same number of ticks are congruent.
• Arcs at angles: angles with the same number of arcs are congruent.

Reading the figure above: $\overline{PQ} \cong \overline{RS}$, $\overline{QR} \cong \overline{SP}$, $\angle P \cong \angle R$, and $\angle Q \cong \angle S$.

Worked examples

Example 1 — Segments from lengths

$AB = 7$ cm and $CD = 7$ cm. Are the segments congruent? Write a statement.

Segments are congruent when their lengths are equal, and $7 = 7$.

Answer: Yes. $\overline{AB} \cong \overline{CD}$

Example 2 — Angles from measures

$m\angle P = 55°$ and $m\angle Q = 55°$. Are the angles congruent? Write a statement.

Angles are congruent when their measures are equal.

Answer: Yes. $\angle P \cong \angle Q$

Example 3 — Not congruent

$m\angle A = 40°$ and $m\angle B = 50°$. Determine congruence.

Since $40 \ne 50$, the measures are different.

Answer: Not congruent. $\angle A \ncong \angle B$

Example 4 — Using a congruence statement to find a measurement

Given $\overline{RS} \cong \overline{TU}$ and $RS = 12$ in, find $TU$.

Congruent segments have equal lengths, so $TU = RS$.

Answer: $TU = 12$ in

Example 5 — Splitting a segment into congruent parts

A shelf board $18$ cm long is cut into two congruent pieces. How long is each piece?

Congruent pieces have equal lengths, so each is half the whole.

$$18 \div 2 = 9$$

Answer: $9$ cm each

Guided practice

1. $AB = 9$ cm and $CD = 9$ cm. Are the segments congruent? Write a congruence statement.
2. $m\angle X = 72°$ and $m\angle Y = 72°$. Write a congruence statement.
3. $m\angle G = 88°$ and $m\angle H = 98°$. Are the angles congruent? Explain.
4. Given $\overline{JK} \cong \overline{LM}$ and $JK = 6.5$ in, find $LM$.
5. When should you use the symbol $\cong$, and when should you use $=$?

Independent practice

6. Congruent or not congruent? a) segments of $4$ cm and $4$ cm b) segments of $10$ in and $12$ in c) angles of $35°$ and $35°$ d) angles of $90°$ and $89°$
7. Given $\overline{PQ} \cong \overline{RS}$ and $RS = 21$ mm, find $PQ$.
8. Given $\angle D \cong \angle E$ and $m\angle E = 47°$, find $m\angle D$.
9. Using the marked quadrilateral $PQRS$ in this lesson, list every pair of congruent segments and every pair of congruent angles.
10. A ribbon $25$ cm long is cut into two congruent pieces. Find the length of each piece.
11. Application. A bookcase needs two side boards that are congruent. One board measures $32$ in. The second is cut to $31.5$ in. Are the boards congruent? Explain what has to happen, and state exactly how much material is involved.
12. Reasoning. In a small sketch, $\angle M$ measures $40°$ and its rays are $2$ cm long. In a large poster, $\angle N$ measures $40°$ and its rays are $30$ cm long. Are the angles congruent? Explain what an angle actually measures.

Exit ticket 15.1

1. $AB = 13$ cm and $CD = 13$ cm. Write a congruence statement.
2. Given $\angle R \cong \angle S$ and $m\angle S = 64°$, find $m\angle R$.
3. Are angles of $25°$ and $52°$ congruent? Explain.
4. Explain the difference between $\cong$ and $=$, and give one correct example of each.

Lesson 15.2 — Regular Polygons

Polygons first

A polygon is a closed plane figure made of three or more straight sides that meet only at their endpoints. The corner points where sides meet are vertices (one of them is a vertex).

The number of sides names the polygon.

Two conditions, both required

A regular polygon is a polygon in which all sides are congruent and all angles are congruent.

Both conditions, at the same time. That "and" is the whole lesson, because most figures that fail to be regular fail only one of the two tests.

Here is the angle measure in each regular polygon you will meet in this course. These are facts to use, not something you need to derive.

Notice the naming: a regular triangle is usually called an equilateral triangle, and a regular quadrilateral is called a square. Both are regular polygons; they just have older, shorter names.

Two ways to fail

• A rectangle that is not a square has four congruent angles ($90°$ each) but two different side lengths. It fails the side test, so it is not regular.
• A rhombus that is not a square has four congruent sides but two different angle measures. It fails the angle test, so it is not regular.

Each figure passes one test and fails the other, and one failure is enough.

A square is the one quadrilateral that passes both tests, which is why a square is a regular polygon. And notice that a rectangle whose length and width are equal is a square, so it is regular.

Perimeter of a regular polygon

Because every side of a regular polygon is the same length, its perimeter is a multiplication instead of a long addition. For a regular polygon with $n$ sides of length $s$:

$$P = n \times s$$

That relationship runs both directions: if you know the perimeter and the number of sides, divide to get the side length.

Worked examples

Example 1 — Testing a description

A quadrilateral has four sides of $5$ cm and four angles of $90°$. Is it regular? Name it.

All four sides congruent, all four angles congruent — both tests pass.

Answer: Yes, it is regular. It is a square.

Example 2 — A rectangle

A rectangle measures $5$ in by $8$ in. Is it regular? Explain.

All four angles measure $90°$, so the angle test passes. But the sides are $5$, $8$, $5$, $8$, and $5 \ne 8$, so the side test fails.

Answer: Not regular, because its sides are not all congruent.

Example 3 — A rhombus

A rhombus has four sides of $6$ cm, with angles of $60°$, $120°$, $60°$, and $120°$. Is it regular? Explain.

The side test passes. The angle test fails, since $60 \ne 120$.

Answer: Not regular, because its angles are not all congruent.

Example 4 — Perimeter of a regular hexagon

Find the perimeter of a regular hexagon with sides of $7$ cm.

A hexagon has $6$ sides, all congruent.

$$P = n \times s = 6 \times 7 = 42$$

Answer: $42$ cm

Example 5 — Working backward to a side length

A regular pentagon has a perimeter of $45$ in. Find the length of one side.

A pentagon has $5$ congruent sides, so divide.

$$s = \frac{P}{n} = \frac{45}{5} = 9$$

Answer: $9$ in

Guided practice

1. State the two conditions a polygon must meet to be regular.
2. Is a square a regular polygon? Explain.
3. Is a rectangle measuring $4$ cm by $9$ cm regular? Explain.
4. Find the perimeter of a regular octagon with sides of $5$ cm.
5. A regular decagon has a perimeter of $70$ m. Find the length of one side.

Independent practice

6. Regular or not regular? Give a reason for each. a) an equilateral triangle with $6$ cm sides b) a rhombus with $8$ cm sides and angles of $70°$ and $110°$ c) a square with $3$ in sides d) a rectangle measuring $6$ in by $6$ in
7. Name the regular polygon that has: a) $8$ congruent sides b) $5$ congruent sides c) $10$ congruent sides
8. Find the perimeter of a regular hexagon with sides of $12$ cm.
9. A regular pentagon has a perimeter of $60$ in. Find the length of one side.
10. Each angle of a regular hexagon measures $120°$. Find the sum of all six angle measures.
11. Application. A stop sign is a regular octagon with sides of $12$ in. A shop wraps reflective tape around all of its edges exactly once. How many inches of tape are needed, and how many feet is that?
12. Reasoning. Kai says any polygon with all congruent sides must be regular. Give a specific counterexample with measurements and explain which test his figure fails.

Exit ticket 15.2

1. What two conditions make a polygon regular?
2. Name the polygon with $9$ sides.
3. Find the perimeter of a regular hexagon with sides of $9$ cm.
4. Explain why most rectangles are not regular polygons, and name the one kind that is.

Lesson 15.3 — Lines of Symmetry

Folding a figure onto itself

A line of symmetry is a line that divides a figure into two parts that match exactly when the figure is folded along that line. Because the two parts match exactly, they are congruent parts — same size, same shape.

Folding is a fair test, and it is worth doing with paper at least once. Cut out a paper square, fold it corner to corner, and the two halves land on each other with no overhang. That fold line is a line of symmetry. Now fold the same square along a slanted line that is not through the center and the halves do not match. That line is not a line of symmetry.

Regular polygons are generous with symmetry

Every regular polygon has exactly as many lines of symmetry as it has sides.

The reason is the regularity itself. Since all the sides are congruent and all the angles are congruent, the figure looks the same from every vertex and from every side, so a symmetry line exists in every one of those directions.

Where the lines go depends on odd or even

Odd number of sides. Every line of symmetry runs from one vertex through the midpoint of the opposite side. There is no side directly across from a side, so no line connects two vertices.

Even number of sides. The lines come in two kinds, half of each. Half connect two opposite vertices, and half connect the midpoints of two opposite sides.

So a square has $4$ lines: $2$ diagonals and $2$ through side midpoints. A regular hexagon has $6$ lines: $3$ vertex to vertex and $3$ midpoint to midpoint. For a regular polygon with an even number of sides $n$, there are $\dfrac{n}{2}$ of each kind.

Figures with few or no lines of symmetry

Irregular figures are far less generous, which is another way to see that regularity is doing real work.

• A rectangle that is not a square has only 2 lines of symmetry: the two lines through the midpoints of opposite sides. Its diagonals are not lines of symmetry — fold a $3$ by $7$ rectangle along a diagonal and the halves do not land on each other.
• A parallelogram that is not a rectangle or a rhombus has 0 lines of symmetry.
• A scalene triangle, with three different side lengths, has 0.

Worked examples

Example 1 — Equilateral triangle

How many lines of symmetry does an equilateral triangle have, and where do they go?

It has $3$ sides, so $3$ lines. With an odd number of sides, each line runs from a vertex to the midpoint of the opposite side.

Answer: $3$ lines, each from a vertex through the midpoint of the opposite side

Example 2 — Square

Describe all lines of symmetry of a square.

Four sides means $4$ lines. Four is even, so half are vertex to vertex and half are midpoint to midpoint: $4 \div 2 = 2$ of each.

Answer: $4$ lines — the $2$ diagonals, plus $2$ lines joining midpoints of opposite sides

Example 3 — Regular pentagon

How many lines of symmetry does a regular pentagon have? Does any of them join two vertices?

Five sides means $5$ lines. Five is odd, so each line goes from a vertex to a side midpoint.

Answer: $5$ lines; none join two vertices

Example 4 — Regular hexagon

Describe all lines of symmetry of a regular hexagon.

Six sides means $6$ lines, and $6 \div 2 = 3$ of each kind.

Answer: $6$ lines — $3$ joining opposite vertices and $3$ joining midpoints of opposite sides

Example 5 — A regular 12-gon

A regular polygon has $12$ sides. How many lines of symmetry does it have, and how do they split into the two kinds?

The count matches the number of sides, and $12$ is even.

$$12 \div 2 = 6$$

Answer: $12$ lines — $6$ vertex to vertex and $6$ midpoint to midpoint

Guided practice

1. How many lines of symmetry does an equilateral triangle have?
2. How many lines of symmetry does a square have?
3. How many lines of symmetry does a regular pentagon have?
4. A regular octagon has how many lines of symmetry? How many are vertex to vertex, and how many are midpoint to midpoint?
5. Explain why the two parts made by a line of symmetry are congruent.

Independent practice

6. Give the number of lines of symmetry: a) regular heptagon b) regular nonagon c) regular decagon d) regular polygon with $20$ sides
7. For a regular hexagon, how many lines of symmetry pass through two opposite vertices, and how many pass through the midpoints of opposite sides?
8. In a regular pentagon, does any line of symmetry connect two vertices? Explain using odd and even.
9. A rectangle measures $3$ cm by $7$ cm. How many lines of symmetry does it have? Explain why its diagonals do not count.
10. A regular polygon has $15$ lines of symmetry. How many sides does it have? Are its symmetry lines vertex to vertex, vertex to midpoint, or a mix? Explain.
11. Application. A designer cuts a paper regular hexagon along a line of symmetry to get two matching pieces for a quilt block. Explain why the two pieces are guaranteed to be congruent, and state how many different folds would work.
12. Reasoning. Rosa says the diagonals of any parallelogram are lines of symmetry. Give a specific parallelogram that shows she is wrong, and explain what goes wrong when you fold along the diagonal.

Exit ticket 15.3

1. How many lines of symmetry does a square have?
2. How many lines of symmetry does a regular decagon have?
3. What is always true about the two parts a line of symmetry creates?
4. Explain why a regular polygon has exactly as many lines of symmetry as it has sides.

Lesson 15.4 — Congruent and Noncongruent Polygons

Corresponding parts

Two polygons are congruent when every pair of corresponding sides is congruent and every pair of corresponding angles is congruent.

Corresponding parts are the parts that occupy matching positions in the two figures. The naming order of a congruence statement tells you the matching. In

$$\triangle ABC \cong \triangle DEF$$

the vertices match in order: $A$ with $D$, $B$ with $E$, $C$ with $F$. From that single statement you can read off six facts:

Because the order carries the information, write congruence statements carefully. $\triangle ABC \cong \triangle DEF$ and $\triangle ABC \cong \triangle EFD$ make different claims.

Position does not matter

A congruent copy may be slid, turned, or flipped. It is still congruent, because sliding, turning, and flipping change where a figure sits without changing any length or any angle measure.

The two triangles above sit at different angles on the page, yet all three pairs of corresponding sides are congruent and all three pairs of corresponding angles are congruent. So $\triangle ABC \cong \triangle DEF$.

Noncongruent

Two polygons are noncongruent when at least one pair of corresponding sides or corresponding angles is not congruent. One mismatch is enough. You do not need to check every part once you have found a difference — but you do need to check every part before declaring two figures congruent.

"Same shape" is not the same as "congruent"

This distinction matters, and it is where most errors live.

Both figures above are squares. They have the same shape, and all eight angles measure $90°$, so every pair of corresponding angles is congruent. But $4 \ne 9$, so no pair of corresponding sides is congruent.

Congruent means same shape and same size. Two figures with the same shape but different sizes are called similar, not congruent. Every congruent pair is also similar, but a similar pair is only congruent when the sizes match too.

That is why matching angles alone never settles congruence. All equilateral triangles have three $60°$ angles, no matter how big they are.

A checklist for deciding

1. Do the two polygons have the same number of sides? If not, stop — noncongruent.
2. Pair up corresponding sides and compare their lengths.
3. Pair up corresponding angles and compare their measures.
4. All pairs congruent means congruent. Any pair not congruent means noncongruent.

Worked examples

Example 1 — Two triangles from measurements

Triangle 1 has sides $3$ cm, $4$ cm, $5$ cm with angles of about $37°$, about $53°$, and $90°$. Triangle 2 has sides $3$ cm, $4$ cm, $5$ cm with the same three angle measures, and the parts match up in that order. Congruent or noncongruent?

Compare the corresponding sides: $3 = 3$, $4 = 4$, $5 = 5$. Compare the corresponding angles: each measure equals its partner, including the $90°$ pair. Every pair matches.

Answer: Congruent

Example 2 — Two rectangles

Rectangle $ABCD$ measures $4$ in by $7$ in. Rectangle $EFGH$ measures $5$ in by $7$ in. Congruent or noncongruent?

All eight angles are $90°$, so the angles all match. But one pair of corresponding sides measures $4$ in and $5$ in, and $4 \ne 5$.

Answer: Noncongruent, because a pair of corresponding sides is not congruent

Example 3 — Two squares

A square has sides of $5$ cm and another has sides of $8$ cm. Do they have the same shape? Are they congruent?

Both are squares, so the shape is the same and all angles are $90°$. The side lengths differ.

Answer: Same shape, but noncongruent, because $5 \ne 8$

Example 4 — Using a congruence statement

Given $\triangle ABC \cong \triangle DEF$ with $AB = 6$ m, $BC = 8$ m, and $m\angle A = 50°$, find $DE$, $EF$, and $m\angle D$.

Read the matching from the order of the letters: $A$ with $D$, $B$ with $E$, $C$ with $F$. So $\overline{AB}$ corresponds to $\overline{DE}$, $\overline{BC}$ corresponds to $\overline{EF}$, and $\angle A$ corresponds to $\angle D$.

$$DE = AB = 6 \text{ m}$$ $$EF = BC = 8 \text{ m}$$ $$m\angle D = m\angle A = 50°$$

Answer: $DE = 6$ m, $EF = 8$ m, $m\angle D = 50°$

Example 5 — All angles congruent is not enough

One equilateral triangle has sides of $4$ cm and another has sides of $9$ cm. Every angle in both is $60°$. Congruent or noncongruent?

All three pairs of corresponding angles are congruent. But the corresponding sides measure $4$ cm and $9$ cm.

Answer: Noncongruent. Matching angles fix the shape but not the size, so the sides must match too.

Guided practice

1. Two triangles each have sides $5$ cm, $12$ cm, $13$ cm and angles of about $23°$, about $67°$, and $90°$, matching in that order. Congruent or noncongruent?
2. Two rectangles each measure $3$ ft by $9$ ft. Congruent or noncongruent? Explain.
3. One rectangle measures $3$ ft by $9$ ft and another measures $3$ ft by $10$ ft. Congruent or noncongruent? Name the pair that fails.
4. Given $\triangle PQR \cong \triangle STU$ and $PQ = 14$ cm, find $ST$.
5. Explain why two squares with different side lengths are not congruent even though both are squares.

Independent practice

6. Congruent or noncongruent, with a reason for each: a) two equilateral triangles with $6$ cm sides b) an equilateral triangle with $6$ cm sides and one with $7$ cm sides c) two squares with $10$ in sides d) a square with $10$ in sides and a rhombus with $10$ in sides whose angles measure $80°$ and $100°$
7. Given $ABCD \cong WXYZ$, name the side that corresponds to $\overline{BC}$ and the angle that corresponds to $\angle D$.
8. Given $\triangle ABC \cong \triangle DEF$ with $AB = 9$ m, $BC = 5$ m, $AC = 11$ m, and $m\angle B = 100°$, find $DE$, $EF$, $DF$, and $m\angle E$.
9. A triangle is turned upside down and set beside an identical triangle. Are the two triangles congruent? Explain what turning a figure does and does not change.
10. Two regular hexagons each have sides of $3$ cm. Are they congruent? Explain how the definition of regular makes this quick to decide.
11. Application. A factory stamps metal plates that must be congruent to a template pentagon with sides $4$ cm, $4$ cm, $6$ cm, $6$ cm, $5$ cm and matching angles. One plate comes out with sides $4$ cm, $4$ cm, $6$ cm, $6$ cm, $5.2$ cm and otherwise matching angles. Is the plate congruent to the template? State what has to change and by how much.
12. Reasoning. Explain the difference between "same shape" and "congruent." Use two squares with specific side lengths as your example, and name the word that describes figures with the same shape but different sizes.

Exit ticket 15.4

1. Two rectangles each measure $5$ in by $8$ in, with corresponding parts matching. Congruent or noncongruent?
2. Given $\triangle ABC \cong \triangle XYZ$ and $m\angle C = 35°$, find $m\angle Z$.
3. A square has $4$ cm sides and another has $9$ cm sides. Do they have the same shape? Are they congruent?
4. Explain what must be true about the sides and angles of two polygons for them to be congruent.

Chapter 15 Review

Vocabulary. congruent · noncongruent · similar · polygon · vertex · regular polygon · equilateral triangle · line of symmetry · corresponding sides · corresponding angles

Part A — Identifying regular polygons (6.MG.4a)

1. State the two conditions a polygon must satisfy to be regular.
2. Regular or not regular, with a reason: a) a square with $7$ cm sides b) a rectangle measuring $5$ m by $11$ m c) a rhombus with $9$ in sides and angles of $60°$ and $120°$ d) an equilateral triangle with $2$ ft sides
3. Name the regular polygon with each number of congruent sides: a) $6$ b) $8$ c) $10$
4. Find the perimeter of a regular pentagon with sides of $11$ cm.
5. A regular octagon has a perimeter of $56$ in. Find the length of one side.

Part B — Lines of symmetry in regular polygons (6.MG.4b)

6. Give the number of lines of symmetry: a) equilateral triangle b) square c) regular hexagon d) regular decagon
7. Describe where each line of symmetry of a regular pentagon goes, and explain why none of them joins two vertices.
8. A regular octagon's $8$ lines of symmetry split into two kinds. Describe each kind and give how many there are of each.
9. What is always true about the two parts created by a line of symmetry?
10. A rectangle measures $4$ cm by $10$ cm. Give its number of lines of symmetry and explain why the diagonals are not among them.

Part C — Congruence of segments and angles (6.MG.4c)

11. $AB = 15$ mm and $CD = 15$ mm. Write a congruence statement.
12. Given $\angle J \cong \angle K$ and $m\angle K = 118°$, find $m\angle J$.
13. Are angles of $61°$ and $16°$ congruent? Explain.
14. Explain when to use $\cong$ and when to use $=$, giving one correct example of each.

Part D — Congruent and noncongruent polygons (6.MG.4d)

15. Congruent or noncongruent, with a reason: a) two rectangles, each $6$ cm by $9$ cm b) two rectangles, $6$ cm by $9$ cm and $6$ cm by $8$ cm c) two regular hexagons with $5$ mm sides d) two squares, one with $3$ in sides and one with $12$ in sides
16. Given $\triangle ABC \cong \triangle RST$ with $AB = 12$ cm, $AC = 7$ cm, and $m\angle B = 45°$, find $RS$, $RT$, and $m\angle S$.
17. Given $PQRS \cong TUVW$, name the side corresponding to $\overline{RS}$ and the angle corresponding to $\angle Q$.

Part E — Mixed application and reasoning

18. Application. A tile pattern uses regular hexagons with sides of $4$ cm. Find the perimeter of one tile, state how many lines of symmetry it has, and explain why any two of these tiles are congruent.
19. Application. A woodworker needs two shelf brackets congruent to a template triangle with sides $8$ in, $15$ in, $17$ in. One bracket measures $8$ in, $15$ in, $17$ in and one measures $8$ in, $15$ in, $16$ in. Which bracket is congruent to the template? Explain what is wrong with the other.
20. Reasoning. A classmate says two polygons must be congruent if all their corresponding angles are congruent. Give a counterexample with measurements, explain the error, and name the word that correctly describes the classmate's figures.

Standards coverage check — Chapter 15

Answer keys for every set in this chapter are in Appendix A.